We associate to every function $u\in GBD(\Omega)$ a measure $\mu_u$ with values in the space of  symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for  functions of bounded deformation. We show that this measure $\mu_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega)$ functions. We then characterise the space $GSBD(\Omega)$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega)$ such that $\mu^c_u=0$. This is joint work with Gianni Dal Maso. 

In the talk, we discuss scaling laws for singular perturbation problems of "staircase type" within the geometrically linearized theory of elasticity. More precisely, we focus on a three-well problem and show that the scaling depends both on the lamination order of the prescribed Dirichlet boundary data and on the number of (non-)degenerate symmetrized rank-one directions in the symmetrized lamination convex hull. Our analysis is based on localization techniques in Fourier space. This is joint work with Angkana Rüland. 

We study the $\Gamma$-convergence of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $\Omega \subset \mathbb{R}^n$ and taking values in the unit circle $S^1 \subset \mathbb{R}^2$. Depending on the domain of the functional, two different Gamma-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map $u$ whose measure of the jump set is minimal. The latter requires ad hoc compactness results for sequences of liftings which, besides being interesting by themselves, also allow to deduce existence of a jump minimizing lifting. This is based on a joint work with Giovanni Bellettini and Riccardo Scala. 

In this talk, we will study a modified two-phase Canham-Helfrich energy, proposed as a model for lipid raft formation in biological membranes. The model involves a coupling between membrane curvature and lipid concentration, which can lead to the formation of patterns in suitable parameter regimes. To better understand the minimizers of the proposed energy, we will derive a scaling law for the infimal energy.  A key step in the proof is the development of novel nonlocal and nonlinear interpolation inequalities that bound fractional Sobolev seminorms in terms of the Modica-Mortola energy.  In the second part, we turn to the associated gradient flow equation and prove the existence of weak solutions for both regular and singular potentials. Additionally, we will see numerical simulations that illustrate the emergence of patterns in different parameter regimes. The results are based on joint work with Janusz Ginster and Barbara Zwicknagl, as well as on collaboration with Patrik Knopf and Dennis Trautwein. 

In this talk, I will discuss the stochastic homogenisation of integral functionals with degenerate growth conditions. After a brief overview of recent developments in the literature, I will present new results on the stochastic homogenisation of strongly anisotropic degenerate functionals. In particular, I will consider functionals whose integrands exhibit degenerate growth and coercivity of order $p > 1$, governed by two nonnegative random weight functions: $\Lambda$ (controlling growth) and $\lambda$ (controlling coercivity). A key feature of this setting is that the ratio $\Lambda/\lambda$ is unbounded with positive probability. This talk is based on a joint work with C. Zeppieri. 

Motivated by a model for elastic bio-membranes, we study the minimization of the Helfrich energy among closed surfaces of prescribed area that are trapped inside a container. We show existence of minimizers in the class of immersed bubble trees and derive the Euler-Lagrange equations which involve a measure term that concentrates on the free boundary. This system can be transferred into a system of conservation laws with a Jacobian structure which allows us to conclude the optimal regularity for solutions. This is joint work with M. Röger (Dortmund). 

In this talk, we study the rigidity properties of a differential inclusion in linearized elasticity. We will introduce and discuss a set $K$ of three diagonal strains that are pairwise incompatible but with non-trivial (symmetrized) rank-$1$-convex hull. This is called a $T3$ structure. We first prove that $K$ is rigid at the level of exact solutions, namely that Lipschitz maps whose gradient is locally in $K$ must be affine. After this, we study the scaling behaviour of the corresponding singularly-perturbed elastic energy, giving quantitative information on the flexibility of $K$ in the sense of approximate solutions. This is based on a joint work with R. Indergand, D. Kochmann, A. Rüland, and C. Zillinger. 

In this talk, I will discuss the derivation of a two-dimensional membrane model for hyper-viscoelastic materials with inertia, starting from the full three-dimensional theory. As the derivation of the thin film limit at the level of the equation is prevented by the low regularity of solutions, the hyperbolic minimizing movements scheme is introduced. This scheme, introduced in [Benešová, Kampschulte, Schwarzacher, JEMS 2024], is a two–time scale variational approximation which accounts for the reformulation  of the problem in terms of incremental minimization of a energy-dissipation functional, where inertia is discretized in time. Within this framework, the asymptotic analysis as the thickness of the body vanishes is carried out via a $\Gamma$-convergence argument at the level of the functional and, subsequently, the continuous-in-time dynamics is recovered. This is a joint work with Martin Kružík. 

The Faddeev-Skyrme model is a by now classical nonlinear $O(3)$-sigma model which has proved very successful in quantum field theory, in particular in detecting topological solitons.  A prototype mathematical model for fields $u\colon S^3\to S^2$ consists of the harmonic map energy together with its symplectic variant acting as a singular perturbation. We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev–Skyrme energy in its homotopy class, for a sufficiently large coupling constant. This is joint work with A. Guerra and X. Lamy.